Dynamic Dilution for Polymers with Backbones

With respect to the arm relaxation time, we have to reconsider the role of dynamic dilution when both arms and backbones are present. For a melt of pure monodisperse stars, the Ball-McLeish theory for dynamic dilution predicts that the effective volume fraction of entangling chains decreases towards zero as the arms relax see Section 9.3.2. However, arms of 

Notice that when the volume fraction of arms 0 approaches unity, Eq. 9.18 yields the expression for dynamically diluted star arms, while when 0 approaches zero, Eq. 9.18 reverts to the arm relaxation time in the absence of dynamic dilution. Eq. 9.18 should then be used in Eq. 9.13 to compute the backbone relaxation time. In Eq. 9.10 the number of entanglements Z should be taken to be the number of entanglements of the backbone with other backbones, because the arms have released all their constraints on the time scales over which the backbone is moving. Since the backbone volume fraction is 0, the number of backbone/backbone entanglements is 0(, where is the number of undiluted backbone entangle- 

There is an extra factor of 25/16 = (5/4) in the latter because (as defined by Eq. 6.21) was used rather than,  

A factor of two arising from the definition of = 2 q, whereas in Daniels et al. q includes only the arms attached to one of the two branch points, while in Eq. 9.20 represents the total number of arms attached to the backbone, 

The general value of the dilution exponent ain Daniels et a/., which we could easily include in Eq. 9.20, 

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